A Note on the Expected Time for Finding Maxima by List Algorithms

Maxima in Rd are found incrementally by maintaining a linked list and comparing new elements against the linked list. If the elements are independent and uniformly distributed in the unit square [0, 1]d , then, regardless of how the list is manipulated by an adversary, the expected time is O(n logd-2 n). This should be contrasted with the fact that the expected number of maxima grows as logd-1 n, so no adversary can force an expected complexity of n logd-1 n. Note that the expected complexity is O(n) for d = 2. Conversely, there are list-manipulating adversaries for which the given bound is attained. However, if we naively add maxima to the list without changing the order, then the expected number of element comparisons is n + o(n) for any d 2. In the paper we also derive new tail bounds and moment inequalities for the number of maxima. Key Words. Outer layers, Maxima, List algorithms, Expected time, Randomized algorithms, Probabilistic analysis.

• Type: Journal
• Year: 1999
• Where: ALGORITHMICA
• Authors: Luc Devroye
• Keywords: ALGORITHMICA 1999, Expected Complexity, Expected Number, Linked List